%% DIFFERENTIAL EQUATIONS WITH MATLAB % CHAPTER 5 %% % *An illustration of stability* %% % We look at how the solution to %% % % $$y' = e^{\mbox{--}x} \,\mbox{--} \,2y,\quad y(0)=c$$ % % depends on the initial value c. syms x eqn = 'Dy = exp(-x) - 2*y' Y = dsolve(eqn, 'y(0) = c', 'x') %% % We plot the solution for initial values .9, 1 and 1.1. figure; hold on for j = -1:1 ezplot(subs(Y,'c',1 + j/10),[-1,1]) end axis tight hold off %% % The inital values are close together. (Two consecutive functions have % values which differ by 0.1.) The plots remain close together for % positive x, and in fact become closer together. This equation is stable. % The derivative of the right side with respect to y is negative. %% % *An illustration of instability* %% % Now we look at the same question for the equation %% % % $$y' = e^{\mbox{-}x} + 2y,\quad y(0) = c$$ % eqn2 = 'Dy = exp(-x) + 2*y' Y_2 = dsolve(eqn2,'y(0)=c','x') %% % We plot the solution for inital values -11/30, -10/30 and -9/30. figure; hold on for j = -1:1 ezplot(subs(Y_2,'c',-1/3 + j/30),[-1,1]) end axis tight hold off %% % The functions have initial values at x=0 which are close together. % (Two consecutive functions have initial values which differ by 1/30). % However, the plots spread apart for positve x. This equation is % unstable. The derivative of the right side with respect to y is positive.